An efficient financial portfolio provides the maximum risk adjusted return, which can be the greatest expected return for a given level of risk or the lowest risk for a given expected return. In general, risk can be measured symmetrically or asymmetrically. Symmetric risk measures include variance and standard deviation. Asymmetric risk measures include Value at Risk (VaR), which is a quantile of projected gains or losses over a target horizon, such as described in Jorion, P., “Value at Risk: the New Benchmark for Managing Financial Risk,” p. 22, McGraw Hill (2d ed. 2001), the disclosure of which is incorporated by reference. Risk adjusted return is one specific type of risk measure for gauging portfolio efficiency. Risk adjusted return can be used by investors to compare the performance of portfolio managers and by portfolio managers to rank securities as potential investment candidates. Optimizing risk adjusted return can improve portfolio efficiency. Conventionally, risk factors, including vectors of financial asset returns, can be modeled using a Normal, or Gaussian, probability distribution either implicitly or explicitly during risk adjusted return optimization and measurement.
Although parametric Normal distributions allow for portfolio optimization over one or more time horizons, the returns for most financial asset classes and risk factors are leptokurtic and skewed and Normal distributions cannot account for such heavy tailed and asymmetric behavior. A probability distribution is considered leptokurtic if the distribution exhibits kurtosis, where the mass of the distribution is greater in the tails and is less in the center or body, when compared to a Normal distribution. A number of quantitative measures of kurtosis have been developed, such as described in Scheffe, “The Analysis of Variance,” p. 332, Wiley & Sons, Inc., New York (1959), the disclosure of which is incorporated by reference. In addition, a probability distribution can be considered asymmetric if one side of the distribution is not a mirror image of the distribution, when the distribution is divided at the maximum value point or the mean.
An example of risk adjusted return optimization that implicitly uses parametric Normal distribution assumptions is mean-variance optimization, such as described in Markowitz, H. M., “Portfolio Selection,” Jour. of finan., Vol. 7, No. 1, pp. 77-91 (1952), the disclosure of which is incorporated by reference. Normal distributions fare poorly in risk adjusted return optimization for several reasons. For instance, Normal distribution “tails” fall off too quickly to account for leptokurtic behavior. As well, the returns of many financial assets exhibit skewed distributions. Moreover, the risk adjusted returns and risk factors for many types of financial assets remain leptokurtic and skewed even after removal of clustering of volatility effects through parametric Normal distributions. As a result, Normal distributions inaccurately measure risk adjusted returns and risk factors during financial portfolio optimization.
An alternative method for optimizing risk adjusted returns is described in Rockafellar, R. T. et al., “Optimization of Conditional Value-at-Risk,” Jour. of Risk, Vol. 2, No. 3, pp. 21-41 (2000), the disclosure of which is incorporated by reference. Expected return is maximized as a Linear Programming (LP) problem using expected tail loss (ETL), also known as conditional value at risk (CVaR) or expected shortfall, as a measure of risk for the risk factors. Although. ETL is an asymmetric measure of risk that measures only down side risk, that is, the risk of loss, ETL differs from variance by only a constant and is modeled as a symmetric measure under a Normal distribution. As well, ETL can be determined using an historical method, which attempts to estimate an empirical distribution for the returns and risk factors. However, empirical distributions can only model actual historical events and are inherently limited by the historical record.
The Sharpe ratio, such as described in Sharpe, W. F., “The Sharpe Ratio,” Jour. of Port. Mgt., pp. 49-58 (Fall 1994), the disclosure of which is incorporated by reference, is a classic example of risk adjusted return. The Sharpe ratio can be expressed as the univariate equation:
  ρ  =            E      -              (                  r          -                      r            f                          )                    STD      ⁢                          ⁢              (                  r          -                      r            f                          )            for the risky asset return r, rf is a return of a risk-free asset f, and STD is the standard deviation. The Sharpe ratio assumes that risk can be measured by a standard deviation of the returns relative to the risk free rate.
An alternative ratio that accounts for the asymmetric behavior of financial asset returns is the Sortino-Stachel ratio, such as described in Sortino, F. A., “Upside-Potential Ratios Vary by Investment Style,” Pensions and Invests., Vol. 28, pp. 30-35 (2000), the disclosure of which is incorporated by reference. The Sortino-Stachel ratio attempts to use a measure of down side risk, but fails to address leptokurtic behavior.
An alternative ratio that accounts for the leptokurtic behavior of financial asset returns is the stable ratio, such as described in Ortobelli, S. L. et al., “The Problem of Optimal Portfolio with Stable Distributed Returns,” Working Paper, UCLA (2003), the disclosure of which is incorporated by reference. The stable ratio assumes that excess return, (r−rf), is a stable non-Gaussian distribution. Although addressing leptokurtic behavior, the stable ratio fails to address asymmetric behavior. Further, this ratio uses only an empirical measure of risk, rather than an actual measure of the probability of loss or tail risk, as could be achieved with a parametric distribution, diminishing the potential accuracy.
An alternative ratio that accounts for both leptokurtic and asymmetric behavior is the Farinelli-Tibiletti ratio, such as described in Farinelli, S. et al., “Sharpe Thinking With Asymmetrical Preferences,” Tech. Report, Univ. of Torino (2003), the disclosure of which is incorporated by reference. The Farinelli-Tibiletti ratio uses asymmetric measures of “up-side” and “down-side,”. This ratio uses only an empirical measure of risk, rather than an actual measure of the probability of loss or tail risk, as could be achieved with a parametric distribution, diminishing the potential accuracy.
Therefore, there is a need for an approach to determining an optimization of risk adjusted return of financial assets and portfolios that accounts for leptokurtic and asymmetric effects independent of the limitations of Normal or empirical distributions.
However, in addition to obtaining optimal portfolio allocations on the basis of leptokurtic and asymmetric distributions using proper risk and reward measures, a framework for portfolio optimization can be used to determine optimized portfolio allocations when severe turnover constraints are necessary without the need to solve the optimization problem. These portfolio allocations are based on so-called local properties of the objective of the optimal portfolio problem. From a financial viewpoint, the statistics marginal contribution to risk and implied returns are considered. Although an overview of the classical marginal contribution to risk based on the Markowitz problem as discussed above is described in Grinold, R. et al., “Active Portfolio Management: A Quantitative Approach for Producing Superior Returns and Controlling Risk”, McGraw-Hill (1999), the disclosure of which is incorporated by reference, a different approach to marginal contribution to risk is needed when the financial returns follow leptokurtic and skewed distributions. This approach to risk attribution, called Marginal Contribution to ETL (MCETL), is described in Zhang, Y. et al., “Risk Attributions and Portfolio Performance Measurements,” Journal of Applied Functional Analysis, 4(1), pp. 373-402, (2004), the disclosure of which is incorporated by reference. However, MCETL alone does not take into account the expected return of the portfolio and can form reallocation rules for only the minimum risk portfolio but not for the rest of the possible points on the efficient frontier.
Therefore, there is a further need for proving reallocation and reverse optimization of financial assets and portfolios that accounts for leptokurtic and asymmetric effects consistent with a general scenarios based portfolio optimization framework.